CN110203831B - Global sliding mode control method of bridge crane system - Google Patents

Global sliding mode control method of bridge crane system Download PDF

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CN110203831B
CN110203831B CN201910440496.7A CN201910440496A CN110203831B CN 110203831 B CN110203831 B CN 110203831B CN 201910440496 A CN201910440496 A CN 201910440496A CN 110203831 B CN110203831 B CN 110203831B
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trolley
bridge crane
crane system
control
sliding mode
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CN110203831A (en
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武宪青
徐可心
马淼
柯飂挺
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Zhejiang Sci Tech University ZSTU
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B66—HOISTING; LIFTING; HAULING
    • B66C—CRANES; LOAD-ENGAGING ELEMENTS OR DEVICES FOR CRANES, CAPSTANS, WINCHES, OR TACKLES
    • B66C13/00—Other constructional features or details
    • B66C13/18—Control systems or devices
    • B66C13/22—Control systems or devices for electric drives

Abstract

The invention provides a global sliding mode control method of a bridge crane system, which comprises the following steps: determining a dynamic model; determining a control target; designing a damping signal and transforming a dynamic equation; designing a control law; realizing a control method; the control method provided by the invention can realize trolley positioning control and swing elimination control of the bridge crane system, ensures that the bridge crane system has good robustness in the whole operation process, improves the control efficiency of the bridge crane, reduces the probability of safety problems caused by external interference and uncertain parameters in the actual operation of the bridge crane system, and has good application prospect and economic benefit.

Description

Global sliding mode control method of bridge crane system
Technical Field
The invention belongs to the control technology of an under-actuated system, and mainly relates to a global sliding mode control method of a bridge crane system.
Background
The bridge crane system is used as an important transport tool and is widely applied to various industrial occasions, the main purpose of the control is to utilize the trolley to convey the load to the target position as fast and accurately as possible, namely the positioning control of the trolley, and the effective control of the load swing angle in the transport process is realized, so that the load swing angle is zero when the trolley reaches the target position, namely the swing eliminating control is realized. However, due to the under-actuated characteristic of the bridge crane system, the control input directly acts on the trolley to control the displacement of the trolley, and the load swing angle is indirectly controlled by the movement of the trolley, so that the control difficulty is increased.
Currently, a bridge crane system has been studied by a number of scholars in various aspects. In the document [1], the control problem of the displacement of the trolley is converted into a tracking control problem, and a novel track tracking control method is designed, so that the running and control quantity of the trolley are smoother. In the literature [2], a self-adaptive control method based on dissipation theorem is proposed to estimate the trolley mass, the load mass and the lifting rope length on line, considering the problem that the trolley mass, the load mass and the lifting rope length are difficult to measure accurately in practical situations. The documents [3] and [4] respectively consider the shortest time trajectory planning problem of the bridge crane system and the energy optimization problem of the bridge crane system in the transportation process, and the former also restricts the load swing angle, the speed and the acceleration of the trolley and the like. In addition, positioning and anti-sway control based on LQR [5], limited time positioning control [6], an offline trajectory planning method [7], and the like are studied by students.
In the actual operation process of the bridge crane system, the bridge crane system is very easily influenced by external interference and uncertain parameters, and the control efficiency of the bridge crane is reduced. Because the sliding mode control has good robustness to uncertain parameters and external interference, a plurality of scholars research on the sliding mode control of the bridge crane system, for example, a document [8] designs a robust controller of the bridge crane system by respectively applying the traditional sliding mode technology and the step-by-step sliding mode control technology; for a bridge crane system with variable rope length, a scholars proposes an anti-swing positioning controller [9] and an adaptive sliding mode control [10] based on a dynamic sliding mode. Generally, the system state of sliding mode control is divided into an approach mode and a sliding mode, when the system state is in the sliding mode, the system has good robustness [11] to uncertain parameters and external interference, but the characteristic does not exist in the approach mode, namely, the system does not have the whole-course robustness.
Accordingly, there is a need for improvements in the art.
The references are as follows:
1. sun Ning, Yongyong purity, Chenhe, under-actuated bridge crane sway elimination tracking control [ J ] control theory and application 2015,32(8):326 + 333.
2. Mabocu, Fangyong, Wangyitao, Jiangshiping, under-actuated bridge crane system adaptive control [ J ] control theory and application, 2008,25(6):1105 + 1109.
3.Zhang X B,Fang Y C,Sun N.Minimum-time trajectory planning forunderactuated overhead crane systems with state and control constraints[J].IEEE Transactions on Industrial Electronics,2014,61(12):6915-6925.
4.Sun N,Wu Y M,Chen H,Fang Y C.An energy-optimal solution fortransportation control of cranes with double pendulum dynamics:design andexperiments[J].Mechanical Systems and Signal Processing,2018,102:87-101.
5. Liu Bao Dynasty, Jia hong Yu, Chen Neng, three-dimensional bridge crane positioning and anti-swing control research based on LQR algorithm [ J ] computer measurement and control, 2018,26(6):89-93.
6.Wu X F,He L,Gao H,Qian J Y,Wu X Q.Finite-time tracking control ofunderactuated overhead cranes[C].Proceeding of the 30thChinese Control andDecision Conference,2018.Shenyang,China:3146-3151.
7.Sun N,Fang Y C,Zhang Y D,and Ma B J.A novel kinematic coupling-based trajectory planning method for overhead cranes[J].IEEE/ASMETransactions on Mechatronics,2012,17(1):166-173.
8.Tuan L A,Lee S-G.Sliding mode controls of double-pendulum cranesystems[J].Journal of Mechanical Science and Technology,2013,27(6):1863-1873
9. The bridge crane anti-swing positioning controller based on the dynamic sliding mode structure is designed [ J ] to control engineering, 2013,20: 117-.
10.Tuan L A,Moon S-C,Lee W G,Lee S-G.Adaptive sliding mode control ofoverhead cranes with varying cable length[J].Journal of Mechanical Scienceand Technology,2013,27(3):885-893.
11. Liu jin jade, sliding mode variable structure control [ M ] Qinghua university press, 2005:66-71.
Disclosure of Invention
The invention aims to provide an efficient global sliding mode control method of a bridge crane system.
In order to solve the technical problem, the invention provides a global sliding mode control method of a bridge crane system, which comprises the following steps: the method comprises the following steps:
step 1, determining a dynamic model;
based on the Euler-Lagrange equation, the dynamic equation of the bridge crane system is as follows:
Figure BDA0002071881980000021
Figure BDA0002071881980000022
wherein M and M are the mass of the trolley and the load respectively, the length of the fixed lifting rope is l, and g represents the gravity acceleration; x represents the horizontal displacement of the dolly from the initial position,
Figure BDA0002071881980000031
which is indicative of the speed of the trolley,
Figure BDA00020718819800000314
represents the acceleration of the trolley; theta represents a swing angle of the load,
Figure BDA0002071881980000032
the angular velocity representing the swing angle of the load,
Figure BDA0002071881980000033
an angular acceleration representing a load swing angle; fxRepresents a resultant force acting on the bogie:
Fx＝F-Fr(3)
wherein F represents the driving force of the motor acting on the trolley, FrThe friction force between the trolley and the bridge is expressed by selecting the friction force described by formula 4:
Figure BDA0002071881980000034
wherein, Frox、krx∈R+Representing a friction parameter, mu, between the trolley and the railx∈R+Representing the coefficient of static friction between the trolley and the guide rail.
Step 2, determining a control target;
Figure BDA0002071881980000035
wherein p isdxRepresents the target position of the trolley, and T represents the transposition of a matrix or a vector;
step 3, designing a damping signal and transforming a dynamic equation:
the following damping signals are defined:
Figure BDA0002071881980000036
Figure BDA0002071881980000037
represents a damping signal;
for damping signal
Figure BDA0002071881980000038
Integrating twice with respect to time, one can obtain:
Figure BDA0002071881980000039
Figure BDA00020718819800000310
represents a time integral of the damping signal; x is the number ofsRepresents a quadratic time integral of the damping signal;
based on the above-introduced signal (15), a "virtual" trolley position signal χ and the corresponding error signal ξ and its derivative are defined:
χ＝x-λxs(16)
ξ＝χ-pdx(17)
Figure BDA00020718819800000311
wherein λ ∈ R+Which is indicative of a normal number of the cells,
Figure BDA00020718819800000312
representing the first and second derivatives of error signal ξ,
Figure BDA00020718819800000313
representing the first and second derivatives of the "virtual" trolley position signal χ;
(19) and (20) are obtained from the dynamic equation of the bridge crane system and equation (18):
Figure BDA0002071881980000041
Figure BDA0002071881980000042
wherein m (theta) is an auxiliary function one,
Figure BDA0002071881980000043
for the auxiliary function two, g represents the gravity acceleration, and the specific expression thereof is as follows:
m(θ)＝M+msin2θ (7)
Figure BDA0002071881980000044
step 4, designing a control law:
based on the transformed kinetic equations (19) - (20) and the control target, the sliding mode surface is as follows:
Figure BDA0002071881980000045
s is the sliding mode surface, tau represents the integral variable,
Figure BDA0002071881980000046
to represent
Figure BDA0002071881980000047
The position of the initial time;
auxiliary variable
Figure BDA0002071881980000048
The expression of (a) is:
Figure BDA0002071881980000049
wherein the content of the first and second substances,
Figure BDA00020718819800000415
κδ∈R+is to satisfy
Figure BDA00020718819800000416
Is a Hurwitz polynomialZ represents a complex variable.
Based on the bridge crane system models (19) - (20) and the sliding mode surface (21), the following control method is designed:
Figure BDA00020718819800000410
wherein k is1、k2∈R+Is a positive control gain, sgn (·) is a sign function, · denotes an arbitrary function:
step 5, implementation of control method
And controlling the movement of the trolley according to the control signal (23), thereby controlling the load swing angle and realizing the control target of the bridge crane system.
As an improvement of the global sliding mode control method of the bridge crane system, the invention comprises the following steps:
the rope and the load are always located below the bridge:
Figure BDA00020718819800000411
as a further improvement to the global sliding mode control method of the bridge crane system of the present invention:
the method for obtaining the formula (18) and the formula (19) is as follows:
dividing both sides of the formula (2) by ml to obtain
Figure BDA00020718819800000412
Will be provided with
Figure BDA00020718819800000413
Substituting into formula (1) to obtain
Figure BDA00020718819800000414
Substitute (18) into this
Figure BDA0002071881980000051
And
Figure BDA0002071881980000052
thereby obtaining (19) and (20);
as a further improvement to the global sliding mode control method of the bridge crane system of the present invention:
Figure BDA0002071881980000053
for stability analysis of the invention:
the method is used for proving that under the action of the control method (23), the system state is always kept on s-0, and when the trolley moves to the target position, the load swing angle is zero, so that the control target of the system is realized.
According to the defined slip form surface (21):
Figure BDA0002071881980000054
where s (0) represents the value of the slip-form surface s at the initial time, and the above equation proves that the system state is at s ═ 0 at the initial time.
To prove that the system state always remains at s ═ 0 at any time, the following Lyapunov function is defined:
Figure BDA0002071881980000055
as is understood from equations (24) and (25), V (0) is 0, where V (0) represents the value of V at the initial time. By substituting the above equation (25) for time derivative and the control method (23) of the present invention with equations (18) and (20), the following can be obtained:
Figure BDA0002071881980000056
since m (θ) > 0, there are
Figure BDA0002071881980000057
The system is stable in the Lyapunov sense. As shown in the formula (26), V.gtoreq.0 is a non-increasing function, and V (0) ═ 0, soTo be provided with
Figure BDA0002071881980000058
Where t represents time, based on equations (27) and (25), it is possible to obtain:
Figure BDA0002071881980000059
as can be seen from equations (28) and (24), the system state is always maintained at s ═ 0, i.e., the control method provided by the present invention is a global sliding mode control method.
Substituting formula (18) for formula (28):
Figure BDA0002071881980000061
because of the fact that
Figure BDA0002071881980000062
κδIs to satisfy
Figure BDA0002071881980000063
Being the normal number of the Hurwitz polynomial, ξ is globally asymptotically stable with respect to the equilibrium point, i.e.:
Figure BDA0002071881980000064
based on equation (18), equation (30), and equation (19), it is possible to obtain:
Figure BDA0002071881980000065
for the bridge crane system to which the present invention is directed, in consideration of the actual situation, the following assumptions can be made:
cosθ≈1，sinθ≈θ (32)
from equation (32), equations (31) and (2) can be approximated as follows:
Figure BDA0002071881980000066
Figure BDA0002071881980000067
for equation (33), λ > 0, l > 0, g > 0 are known, as determined by the Laus criterion:
Figure BDA0002071881980000068
from the formula (30), by combining the formulae (15), (16) and (17):
Figure BDA0002071881980000069
further, according to equations (35) and (36), it is possible to obtain:
Figure BDA00020718819800000610
in combination with equation (32), equation (35), and equation (37), the following conclusion is reached, taking into account the zero initial condition:
Figure BDA00020718819800000611
combining formula (16), formula (17), formula (30), formula (38), to obtain:
Figure BDA00020718819800000612
therefore, as can be seen from equations (39) and (35), the overhead traveling crane system achieves the system control object under the proposed control method.
In summary, the control method provided by the present invention not only can keep the system state at s ═ 0 all the time, i.e. a global sliding mode control method, but also can realize trolley positioning control and swing elimination control.
The overall sliding mode control method of the bridge crane system has the technical advantages that:
aiming at a bridge crane system, the invention provides a global sliding mode control method. Compared with a general sliding mode control method, the control method provided by the invention has the advantages that the trolley positioning control and the swing elimination control of the bridge crane system are realized, meanwhile, the bridge crane system is ensured to have good robustness in the whole operation process, the control efficiency of the bridge crane is improved, the probability of safety problems caused by external interference and uncertain parameters in the actual operation of the bridge crane system is reduced, and the control method has good application prospect and economic benefit.
Drawings
The following describes embodiments of the present invention in further detail with reference to the accompanying drawings.
FIG. 1 is a model of a bridge crane system to which the present invention is directed;
FIG. 2 is a diagram of simulation results of the control method of the present invention under a zero initial condition;
FIG. 3 is a diagram of simulation results of the control method of the present invention under uncertain parameters;
fig. 4 is a simulation result diagram of the control method of the present invention under external interference.
Detailed Description
The invention will be further described with reference to specific examples, but the scope of the invention is not limited thereto.
Embodiment 1, a global sliding mode control method of a bridge crane system, comprising the steps of:
step 1, determining a dynamic model;
the invention concerns a bridge crane system comprising a trolley capable of one-dimensional translational movement along a bridge and a load connected to the trolley by a fixed rope length. Based on the Euler-Lagrange equation, the dynamic equation of the bridge crane system is as follows:
Figure BDA0002071881980000071
Figure BDA0002071881980000072
wherein M and M are the mass of the trolley and the load respectively, the length of the fixed lifting rope is l, and g represents the gravity acceleration; x represents the horizontal displacement of the dolly from the initial position,
Figure BDA0002071881980000073
which is indicative of the speed of the trolley,
Figure BDA0002071881980000074
represents the acceleration of the trolley; theta represents a swing angle of the load,
Figure BDA0002071881980000075
the angular velocity representing the swing angle of the load,
Figure BDA0002071881980000076
an angular acceleration representing a load swing angle; fxRepresents a resultant force acting on the bogie:
Fx＝F-Fr(3)
wherein F represents the driving force of the motor acting on the trolley, FrIndicating the friction between the trolley and the bridge, based on document [7]](Sun N,Fang Y C,Zhang Y D,and Ma B J.A novel kinematic coupling-basedtrajectory planning method for overhead cranes[J]IEEE/ASME Transactions on mechanics, 2012,17(1): 166-:
Figure BDA0002071881980000077
wherein, Frox、krx∈R+Representing a friction parameter, mu, between the trolley and the railx∈R+Representing the coefficient of static friction between the trolley and the guide rail.
For the convenience of the following description and design of the control method, now dividing both sides of equation (2) by ml, we get the following equation:
Figure BDA0002071881980000081
substituting formula (5) for formula (1) to obtain:
Figure BDA0002071881980000082
wherein the auxiliary function one m (theta) and the auxiliary function two
Figure BDA0002071881980000083
As follows:
m(θ)＝M+msinθ (7)
Figure BDA0002071881980000084
step 2, determining a control target;
the invention is directed to a bridge crane system, the main control objectives of which are: the motion of platform truck is carried the load to target location through the platform truck by the motion of motor direct control platform truck to through the motion indirect control load swing of platform truck, when making the platform truck reach the target location, the load swing is eliminated to zero, promptly:
Figure BDA0002071881980000085
wherein p isdxRepresenting the target position of the trolley and T representing the transpose of the matrix or vector.
Furthermore, considering the actual situation of the bridge crane system movement, the present invention assumes that the lifting rope and load are always located under the bridge, i.e.:
Figure BDA0002071881980000086
step 3, designing damping signals and transforming dynamic equations
To achieve the control objective of eliminating the load swing proposed in step 2, consider the following function:
Figure BDA0002071881980000087
Vθis the load swing.
The first derivative with respect to time of equation (11) is taken in conjunction with equation (5) to obtain:
Figure BDA0002071881980000088
Figure BDA0002071881980000089
the load swing after the first derivative of time.
Based on the above formula, a damping signal satisfying the following relationship is designed
Figure BDA00020718819800000810
The load swing can be effectively restrained:
Figure BDA0002071881980000091
to eliminate the load swing, the following damping signal is defined:
Figure BDA0002071881980000092
Figure BDA0002071881980000093
represents a damping signal;
for the damping signal defined by the above formula
Figure BDA0002071881980000094
Integrating twice with respect to time, one can obtain:
Figure BDA0002071881980000095
Figure BDA0002071881980000096
representing the time product of the damping signalDividing; x is the number ofsRepresents a quadratic time integral of the damping signal;
based on the above-introduced signal (15), the following "virtual" trolley position signal χ and the corresponding error signal ξ and its derivative are defined:
χ＝x-λxs(16)
ξ＝χ-pdx(17)
Figure BDA0002071881980000097
wherein λ ∈ R+Which is indicative of a normal number of the cells,
Figure BDA0002071881980000098
representing the first and second derivatives of error signal ξ,
Figure BDA0002071881980000099
representing the first and second derivatives of the "virtual" trolley position signal χ. Carrying out a series of conversion on a dynamic equation of the bridge crane system to obtain (19) and (20);
the transformation process of the kinetic equation is as follows: dividing both sides of the formula (2) by ml to obtain
Figure BDA00020718819800000910
Substituting this formula into (1) to obtain
Figure BDA00020718819800000911
Substituting (18) into the two equations
Figure BDA00020718819800000912
Figure BDA00020718819800000913
To obtain (19) and (20);
the kinetic equations that can be transformed are shown below:
Figure BDA00020718819800000914
Figure BDA00020718819800000915
wherein m (theta) is an auxiliary function one,
Figure BDA00020718819800000916
for the auxiliary function two, g represents the gravitational acceleration.
Step 4, designing a control law
Based on the transformed kinetic equations (19) - (20) and the control target, the following sliding mode is designed:
Figure BDA00020718819800000917
s is the sliding mode surface, tau represents the integral variable,
Figure BDA00020718819800000918
to represent
Figure BDA00020718819800000919
The position of the initial time;
auxiliary variable
Figure BDA00020718819800000920
The expression of (a) is as follows:
Figure BDA00020718819800000921
wherein the content of the first and second substances,
Figure BDA0002071881980000101
κδ∈R+is to satisfy
Figure BDA0002071881980000102
Is the normal number of the Hurwitz polynomial and z represents the complex variable.
Based on the bridge crane system models (19) - (20) and the sliding mode surface (21), the following control method is designed:
Figure BDA0002071881980000103
wherein k is1、k2E R + is a positive control gain, sgn (-) is a sign function, and represents an arbitrary function:
Figure BDA0002071881980000104
step 5, implementation of control method
The displacement x and the speed of the trolley are measured in real time by using a sensor installed on a bridge crane system
Figure BDA0002071881980000107
Swing angle theta and speed of swing angle of load
Figure BDA0002071881980000106
And measuring, controlling the movement of the trolley according to the control signal (23), indirectly controlling the load swing angle, and realizing the control target of the bridge crane system.
Description of simulation results:
in order to test the control performance of the control method provided by the invention, the part is subjected to two-part simulation experiments, the first part (simulation 1) is the trolley positioning control under the zero initial condition, and the second part (simulation 2) is the robustness test of uncertain parameters and external interference.
Simulation 1, trolley positioning control of zero initial conditions
The parameters of the simulation are selected as follows: m7 kg, M1.025 kg, l 0.6M, g 9.8M/s2The target position of the trolley is as follows: p is a radical ofdx0.6m, the coefficient of friction is: frox＝4.4，μx＝0.01，krx-0.5, the parameters of the control method (23) are selected as:
Figure BDA0002071881980000105
κδ＝2.4，k1＝1.1，k2＝0.2。
the simulation results are shown in fig. 2. As can be seen from FIG. 2, the amplitude of the load swing angle is about 0.05rad during the transportation of the trolley, and the load swing angle has almost no residual swing after the trolley reaches the target position in 7 seconds, which proves that the control method provided by the invention has excellent trolley positioning control performance and swing elimination performance.
Simulation 2, robustness testing
And then, the robustness of the control method provided by the invention to uncertain system parameters and external interference is further detected.
Simulation 2.1 robustness testing for uncertain parameters
The load mass m is changed to 2.025kg and the sling length l is changed to 0.1m, and other parameters are kept unchanged from the simulation 1. The simulation result is shown in fig. 3, and comparing fig. 2 and fig. 3, it can be known that changing the load mass and the length of the lifting rope has little influence on the control performance of the control method provided by the present invention.
Simulation 2.2 robustness test against external disturbances
The system parameter selection is the same as simulation 1, a random disturbance with the amplitude of 1 is added to the load between 4 seconds and 6 seconds, and the simulation result is shown in fig. 4. As can be seen from fig. 4, under the action of the control method provided by the present invention, the external disturbance on the load is quickly suppressed and eliminated, the trolley finally reaches the target position, and the load swing angle is also eliminated to zero.
As can be seen from the results of the simulation 2.1 and the simulation 2.2, the control method provided by the invention has robustness to uncertain parameters and external interference.
Finally, it is also noted that the above-mentioned lists merely illustrate a few specific embodiments of the invention. It is obvious that the invention is not limited to the above embodiments, but that many variations are possible. All modifications which can be derived or suggested by a person skilled in the art from the disclosure of the present invention are to be considered within the scope of the invention.

Claims (4)

1. The global sliding mode control method of the bridge crane system is characterized by comprising the following steps: the method comprises the following steps:
step 1, determining a dynamic model;
based on the Euler-Lagrange equation, the dynamic equation of the bridge crane system is as follows:
Figure FDA0002392159870000011
Figure FDA0002392159870000012
wherein M and M are the mass of the trolley and the load respectively, the length of the fixed lifting rope is l, and g represents the gravity acceleration; x represents the horizontal displacement of the dolly from the initial position,
Figure FDA0002392159870000013
which is indicative of the speed of the trolley,
Figure FDA0002392159870000014
represents the acceleration of the trolley; theta represents a swing angle of the load,
Figure FDA0002392159870000015
the angular velocity representing the swing angle of the load,
Figure FDA0002392159870000016
an angular acceleration representing a load swing angle; fxRepresents a resultant force acting on the bogie:
Fx＝F-Fr(3)
wherein F represents the driving force of the motor acting on the trolley, FrThe friction force between the trolley and the bridge is expressed by selecting the formula (4) as the description of the friction force:
Figure FDA0002392159870000017
wherein, Frox、krx∈R+Representing a friction parameter, mu, between the trolley and the railx∈R+Representing the static friction coefficient between the trolley and the guide rail;
step 2, determining a control target;
Figure FDA0002392159870000018
wherein p isdxRepresents the target position of the trolley, and T represents the transposition of a matrix or a vector;
step 3, designing a damping signal and transforming a dynamic equation:
the following damping signals are defined:
Figure FDA0002392159870000019
Figure FDA00023921598700000110
represents a damping signal;
for damping signal
Figure FDA00023921598700000111
Integrating twice with respect to time, one can obtain:
Figure FDA00023921598700000112
Figure FDA00023921598700000113
represents a time integral of the damping signal; x is the number ofsRepresents a quadratic time integral of the damping signal;
based on the signals introduced by equation (15), a "virtual" trolley position signal χ and the corresponding error signal ξ and its derivatives are defined:
χ＝x-λxs(16)
ξ＝χ-pdx(17)
Figure FDA0002392159870000021
wherein λ ∈ R+Which is indicative of a normal number of the cells,
Figure FDA0002392159870000022
representing the first and second derivatives of error signal ξ,
Figure FDA0002392159870000023
representing the first and second derivatives of the "virtual" trolley position signal χ;
(19) and (20) are obtained from the dynamic equation of the bridge crane system and equation (18):
Figure FDA0002392159870000024
Figure FDA0002392159870000025
wherein m (theta) is an auxiliary function one,
Figure FDA0002392159870000026
for the auxiliary function two, g represents the gravity acceleration, and the specific expression thereof is as follows:
m(θ)＝M+msin2θ (7)
Figure FDA0002392159870000027
step 4, designing a control law:
based on the transformed kinetic equations (19) - (20) and the control target, the sliding mode surface is as follows:
Figure FDA0002392159870000028
s is the sliding mode surface, tau represents the integral variable,
Figure FDA0002392159870000029
to represent
Figure FDA00023921598700000210
The position of the initial time;
auxiliary variable
Figure FDA00023921598700000211
The expression of (a) is:
Figure FDA00023921598700000212
wherein the content of the first and second substances,
Figure FDA00023921598700000214
is to satisfy
Figure FDA00023921598700000215
Is the normal number of the Hurwitz polynomial, z represents the complex variable;
based on the bridge crane system models (19) - (20) and the sliding mode surface (21), the following control method is designed:
Figure FDA00023921598700000213
wherein k is1、k2∈R+Is a positive control gain, sgn (·) is a sign function, · denotes an arbitrary function:
step 5, implementation of control method
And controlling the movement of the trolley according to the control signal (23), thereby controlling the load swing angle and realizing the control target of the bridge crane system.
2. The global sliding mode control method of a bridge crane system according to claim 1, characterized in that:
the rope and the load are always located below the bridge:
Figure FDA0002392159870000031
3. the global sliding mode control method of a bridge crane system according to claim 2, characterized in that:
the method for obtaining the formula (18) and the formula (19) is as follows:
dividing both sides of the formula (2) by ml to obtain
Figure FDA0002392159870000032
Will be provided with
Figure FDA0002392159870000033
Substituting into formula (1) to obtain
Figure FDA0002392159870000034
Substitute (18) into this
Figure FDA0002392159870000035
And
Figure FDA0002392159870000036
thus, (19) and (20) were obtained.
4. The global sliding mode control method of a bridge crane system according to claim 3, characterized in that:
Figure FDA0002392159870000037
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